Theorem bd0 15470

Description: A formula equivalent to a bounded one is bounded. See also bd0r 15471. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 15459	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 15458

This theorem was proved from axioms:  ax-mp 5  ax-bd0 15459

This theorem is referenced by:  bd0r  15471  bdth  15477  bdnth  15480  bdnthALT  15481  bdph  15496  bdsbc  15504  bdsnss  15519  bdcint  15523  bdeqsuc  15527  bdcriota  15529  bj-axun2  15561